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A046951 a(n) is the number of squares dividing n. 107
1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Rediscovered by the HR automatic theory formation program.

a(n) depends only on prime signature of n (cf. A025487, A046523). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).

First differences of A013936. Average value tends towards Pi^2/6 = 1.644934... (A013661, A013679). - Henry Bottomley, Aug 16 2001

We have a(n) = A159631(n) for all n < 125, but a(125) = 2 < 3 = A159631(125). - Steven Finch, Apr 22 2009

Number of 2-generated Abelian groups of order n, if n > 1. - Álvar Ibeas, Dec 22 2014 [In other words, number of order-n abelian groups with rank <= 2. Proof: let b(n) be such number. A finite abelian group is the inner direct product of all Sylow-p subgroups, so {b(n)} is multiplicative. Obviously b(p^e) = floor(e/2)+1 (corresponding to the groups C_(p^r) X C_(p^(e-r)) for 0 <= r <= floor(e/2)), hence b(n) = a(n) for all n. - Jianing Song, Nov 05 2022]

Number of ways of writing n = r*s such that r|s. - Eric M. Schmidt, Jan 08 2015

The number of divisors of the square root of the largest square dividing n. - Amiram Eldar, Jul 07 2020

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Antonio Amariti, Claudius Klare, Domenico Orlando and Susanne Reffert, The M-theory origin of global properties of gauge theories, Nuclear Physics B, Vol. 901 (2015), pp. 318-337, arXiv preprint, arXiv:1507.04743 [hep-th], 2015 (see (A.13)).

Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.

Simon Colton, HR - Automatic Theory Formation in Pure Mathematics

Ian G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34. See delta(n).

Andrew V. Lelechenko, Average number of squares dividing mn, arXiv preprint arXiv:1407.1222 [math.NT], 2014.

Werner Georg Nowak and László Tóth, On the average number of subgroups of the group Z_m X Z_n, International Journal of Number Theory, Vol. 10, No. 2 (2014), pp. 363-374, arXiv preprint, arXiv:1307.1414 [math.NT], 2013.

N. J. A. Sloane, Transforms.

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(p^k) = A008619(k) = [k/2] + 1. a(A002110(n)) = 1 for all n. (This is true for any squarefree number, A005117). - Original notes clarified by Antti Karttunen, Nov 14 2016

a(n) = |{(i, j) : i*j = n AND i|j}| = |{(i, j) : i*j^2 = n}|. Also tau(A000188(n)), where tau = A000005.

Multiplicative with p^e --> floor(e/2) + 1, p prime. - Reinhard Zumkeller, May 20 2007

a(A130279(n)) = n and a(m) <> n for m < A130279(n); A008966(n)=0^(a(n) - 1). - Reinhard Zumkeller, May 20 2007

Inverse Moebius transform of characteristic function of squares (A010052). Dirichlet g.f.: zeta(s)*zeta(2s).

G.f.: Sum_{k > 0} x^(k^2)/(1 - x^(k^2)). - Vladeta Jovovic, Dec 13 2002

a(n) = Sum_{k=1..A000005(n)} A010052(A027750(n,k)). - Reinhard Zumkeller, Dec 16 2013

a(n) = Sum_{k = 1..n} ( floor(n/k^2) - floor((n-1)/k^2) ). - Peter Bala, Feb 17 2014

From Antti Karttunen, Nov 14 2016: (Start)

a(1) = 1; for n > 1, a(n) = A008619(A007814(n)) * a(A064989(n)).

a(n) = A278161(A156552(n)).

(End)

G.f.: Sum_{k>0}(theta(q^k)-1)/2, where theta(q)=1+2q+2q^4+2q^9+2q^16+... - Mamuka Jibladze, Dec 04 2016

From Antti Karttunen, Nov 12 2017: (Start)

a(n) = A000005(n) - A056595(n).

a(n) = 1 + A071325(n).

a(n) = 1 + A001222(A293515(n)).

(End)

L.g.f.: -log(Product_{k>=1} (1 - x^(k^2))^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018

a(n) = Sum_{d|n} A000005(d) * A008836(n/d). - Torlach Rush, Jan 21 2020

a(n) = A000005(sqrt(A008833(n))). - Amiram Eldar, Jul 07 2020

EXAMPLE

a(16) = 3 because 1*16 = 16 and 1|16, 2*8 = 16 and 2|8, 4*4 = 16 and 4|4.

G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + x^10 + ...

MAPLE

A046951 := proc(n)

local a, s;

a := 1 ;

for p in ifactors(n)[2] do

a := a*(1+floor(op(2, p)/2)) ;

end do:

a ;

end proc: # R. J. Mathar, Sep 17 2012

MATHEMATICA

a[n_] := Length[ Select[ Divisors[n], IntegerQ[Sqrt[#]]& ] ]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 26 2012 *)

Table[Length[Intersection[Divisors[n], Range[10]^2]], {n, 100}] (* Alonso del Arte, Dec 10 2012 *)

a[ n_] := If[ n < 1, 0, Sum[ Mod[ DivisorSigma[ 0, d], 2], {d, Divisors @ n}]]; (* Michael Somos, Jun 13 2014 *)

a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (Quotient[ #[[2]], 2] + 1 & /@ FactorInteger @ n)]; (* Michael Somos, Jun 13 2014 *)

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / (1 - x^k^2), {k, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Jun 13 2014 *)

f[p_, e_] := 1 + Floor[e/2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

PROG

(PARI) a(n)=my(f=factor(n)); for(i=1, #f[, 1], f[i, 2]\=2); numdiv(factorback(f)) \\ Charles R Greathouse IV, Dec 11 2012

(PARI) a(n) = direuler(p=2, n, 1/((1-X^2)*(1-X)))[n]; \\ Michel Marcus, Mar 08 2015

(PARI) a(n)=factorback(apply(e->e\2+1, factor(n)[, 2])) \\ Charles R Greathouse IV, Sep 17 2015

(Haskell)

a046951 = sum . map a010052 . a027750_row

-- Reinhard Zumkeller, Dec 16 2013

(Scheme)

(definec (A046951 n) (if (= 1 n) 1 (* (A008619 (A007814 n)) (A046951 (A064989 n)))))

(define (A008619 n) (+ 1 (/ (- n (modulo n 2)) 2)))

;; Antti Karttunen, Nov 14 2016

(Magma) [#[d: d in Divisors(n)|IsSquare(d)]:n in [1..120]]; // Marius A. Burtea, Jan 21 2020

CROSSREFS

Cf. A000005, A000188, A004101, A005117 (positions of 1's), A008619, A008833, A038538, A046952, A052304, A056595, A159631, A007814, A010052, A027750, A239930, A007862, A046523, A064989, A065704, A156552, A278161.

One more than A071325.

Differs from A096309 for the first time at n=32, where a(32) = 3, while A096309(32) = 2 (and also A185102(32) = 2).

Sum of the k-th powers of the square divisors of n for k=0..10: this sequence (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: this sequence (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A082293 (a(n)==2), A082294 (a(n)==3).

Sequence in context: A185102 A049419 A299090 * A159631 A335428 A050377

Adjacent sequences: A046948 A046949 A046950 * A046952 A046953 A046954

KEYWORD

nice,nonn,mult

AUTHOR

Simon Colton (simonco(AT)cs.york.ac.uk)

EXTENSIONS

Data section filled up to 125 terms and wrong claim deleted from Crossrefs section by Antti Karttunen, Nov 14 2016

STATUS

approved

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Last modified December 10 01:29 EST 2022. Contains 358711 sequences. (Running on oeis4.)